Calculations

Standard Atmosphere data. Rather old (1962), but the atmosphere really doesn't change all that much....or that quickly

View attachment Std Atmosph 1962.pdf

More physics equations for change in velocity over a time interval with constant acceleration:

(VelocityFinal) = (VelocityInitial) + (Accel) x (Time)

where units for velocity are meters / second
accel is expressed in meters / (second-squared)
and time is in seconds

This gives us a new value for the increased, final velocity at the end of each time interval. As we work our way down the spreadsheet through all the time intervals and keep adding in all those velocity increments we get to a maximum velocity for the rocket’s entire flight at the end of the motor burn.

Obviously a rocket under thrust has a continuously varying acceleration, but to avoid a whole bunch of calculus (which is a very good thing) I have used a simplified model using the trapezoidal rule and using small time increments (especially during an event like the motor thrust spike where such severe variations occur). Across each 1/100th of a second the final velocity (think of it as “final” for that time increment) is equal to the initial velocity (which is the final velocity from the previous time increment) plus the delta-V (change in velocity) which is accel x time (which is the duration of the time interval).

The value for delta-V in Column M) is determined by multiplying accel (in Column L) by time (in Column P)

The new total velocity (in column K) comes from adding the previous total velocity (from the line above) to the delta-V

Now that a velocity has been estimated for each time interval it is possible to use this info to estimate the aerodynamic drag force acting on the rocket. Drag force is written in many aero textbooks as:

Drag = (q) x (S) x (cd)

where q is dynamic pressure, calculated as = (1/2) x (air density) x (velocity-squared) and has units of Newtons / (meter-squared)
and S is the frontal or cross-sectional area of the flying object against which the air will scrub and cause drag, with units of meters-squared
cd is the drag coefficient and varies depending on shape and smoothness (cd is a dimensionless, “normalized” measure)

Air density data can be difficult to find. A quick search at Wiki will give at least one ISA value for sea level, 15 degrees C as (approximately) 1.22521 kg/m3
https://en.wikipedia.org/wiki/Density_of_air
This agrees with the value shown in my old table (1962 NASA standard atmosphere) for sea level, standard day conditions where density (Greek letter “rho” ) is 0.0023769 slugs/ft3 (conversion using 32.174 pounds per slug, 2.2046 pounds per kilograms, and 3.2808 feet per meter gives a value of 1.225 kg/m3; close enough).

Air density DOES change with altitude (check the standard atmosphere table) but as I stated before a model rocket only reaches a few hundred feet and air density does not change much across this range. I am going to use a constant value, as shown up on line 12.

Dynamic pressure (q) is going to change slightly as the velocity changes for each time increment. A new value is calculated on each line of the flight history and used to estimate a new value of aerodynamic drag for that same line.

So dynamic pressure ( in Column U) is determined by multiplying air density (Row 12, Column D) times velocity squared (Column K) and dividing by 2.

Aerodynamic drag at each line of the flight history is determined by multiplying dynamic pressure (Column U) times the rocket’s cross-section area (Row 16, Column D) times the drag coefficient (Column R)

With values entered for aerodynamic drag, our estimate of forces acting on the rocket is complete (thrust, drag, and weight) and the acceleration data (Column L) will now show a finished set of data. According to this estimate of model rocket performance, my design undergoes some 44 gs of acceleration at the moment of peak thrust. (Let’s see those high-power guys top THAT!)

View attachment 11_delta-v_and_velocity_step_06.xlsx

The next few data items are pretty easy to calculate. We are going to add in the distance traveled during each time increment and the total distance traveled since ignition. Velocity during each time interval is assumed to be constant. Of course the accelerating rocket has a continuously changing velocity, but I am still trying to avoid that calculus stuff by using the trapezoidal approximation along with small time increments.

(Distance traveled) = (Velocity) x (Time)

where distance is in meters
velocity is in meters per second
and time is in seconds

(Total distance traveled) = (Previous distance traveled) + (Distance traveled per time increment)

Just for fun, I squeezed in an extra spreadsheet column (P) with total distance traveled converted to feet so it is a little more recognizable to some folks.

We can begin to see some interesting bits of information appearing on the spreadsheet. The model shows that the rocket sits motionless on the pad for 0.02 seconds after motor ignition before any motion begins. If we are using a short 30-inch Estes launch rod, the rocket covers this distance and separates from the rod with a velocity of 18 meters per second (around 60 feet per second). If we are using a three-foot long launch rod (like the wire from the hobby or hardware stores) the rocket will be traveling 22 meters per second (around 70 feet per second) when it clears the launch rod. This gives a general indication that (assuming little or no rod whip) this rocket should leave the launcher with sufficient airspeed for fin aerodynamic effectiveness and would be reasonably safe to launch in moderate crosswinds.

The rocket reaches maximum thrust at a point 1.5 meters above the launcher (approximately 5 feet). This coincides with the highest internal combustion chamber pressures, and if any of the motors are going to pop a nozzle or do anything else crazy it will probably occur right here. In your face, literally at eyeball level, with seven times the probability of failure. Might be a good idea to use longer launch leads so everyone can be separated a bit further from the pad. (You may want this anyway to get a better view of what will be a fast launch.)

Assuming all motors fire together, the thrust spike for these A10T motors will over while this rocket is still only three meters (16 feet) in the air. The sustainer portion of the motor burn will take over at that point and the rocket will continue accelerating to around 57 m/sec of velocity (190-ish ft/sec) at about 34 meters of altitude. Then it will start coasting.

View attachment 12_distance traveled_step 07.xlsx

At the end of the propellant burn an upper stage motor starts burning the delay charge and the rocket coasts. The delay charge produces minimal thrust, if any at all. Some people claim that the aerodynamic drag increases at this point because the exhaust plume is no longer gushing out to fill in the wake region as the rocket continues moving. Other people claim that the emissions from the delay charge (smoke and gaseous combustion by-products) still serve the purpose of filling the base area behind the rocket, and aero drag does not really increase. I don’t know exactly which is correct. Both arguments make sense to me. I am going to assume for this simulation that the aero drag does not change. If you would like to write your own spreadsheet to reflect some amount of increased drag (+25%? +50%?) it is easy to program this calculation into the spreadsheet.

Motor thrust drops to zero, aero drag either remains constant or changes (if you prefer), and gravity pretty much remains constant. (Again, flights to extreme heights should account for gravity variation with altitude.) The spreadsheet calculation for acceleration (Column L) will continue to work OK if a zero value is inserted for thrust, or if spreadsheet mathematics using zeroes make you nervous you can reprogram the accel calculation at this line and remove the thrust term.

The flight history was modeled with short time increments during the motor burn mainly because this motor has a very quick thrust peak (which I wanted to model in detail) and because there are several events in this part of the flight profile that I wanted to check. Coasting flight is a bit different, not much is going on, and I do not think I need to continue tracking so closely. I opened up the selected time interval to 1/10th of a second but I also could have continued the simulation using the previous 1/100th second if I wanted to check some other flight event.

EXCEL spreadsheets have the wonderful feature where you can highlight a line (including any calculations programmed into individual cells), click copy, drag the mouse down the page to highlight a big new block of lines, then click paste, and you’re done programming a whole block of extended spreadsheet. Add as many lines as you like to model the coasting flight for the time delay of the motor you are simulating. I added lines to the bottom of the spreadsheet from the 0.85-second point (theoretical motor burnout) to the 3.80-second point to represent the three second delay time of the Estes A10-3T motors. Just for grins, I added a line to allow for another tenth of a second (my own SOTP estimate) to represent ejection charge ignition and burn and for recovery deployment.

The simulation shown does not include any further rocket weight changes after motor burnout. Simulation accuracy could be tweaked a bit by accounting for the expended weight of the delay charge during the coasting phase of flight but I doubt that this would change the simulation results very much. This also gets into more motor trivia such as how much weight difference are we even talking about, and whether the delay material is mostly ejected or how much residue gets left inside the motor case afterward?

Triple bonus points to anyone who adjusts the simulation to account for the weight of expended delay materials from the motor and posts the results here.

This simulation tells me that my design will reach 400 feet, and ejection/deployment will occur with a residual (forward) velocity of around 9 m/sec (30-ish ft/sec). This should make fairly benign conditions for recovery deployment without risking much of a zipper or other rough treatment. A four-second delay might have been just about perfect, but you can’t have everything. Of course, at any slightly earlier point in the flight profile (remember that Estes “discount” on delay burn time?) conditions will be a little worse.

With the spreadsheet complete to this point it is also possible to play some “what if” games with the input data. If you were concerned whether you had unfairly penalized your simulation with a drag coefficient that was too high, a user can easily dial back this input at the top (Line 26), let the new value ripple down through the spreadsheet, and quickly see how much change had occurred in accel/velocity/altitude. In this spreadsheet, dropping the value of cd by 33% (down to 0.50) does indeed change the final altitude, but not by an amount that would suggest total and absolute mission disaster.

A user can dial up the input value for rocket weight and also quickly see how this affects the simulation. In general, this will affect rocket performance far more severely than changes in drag coefficient. In this spreadsheet, increasing the rocket weight by 33% (to 280 grams) changes the final altitude quite significantly.

View attachment 13_coast phase_step 8.xlsx

This part is going to be very different for everyone, because everybody has a different idea of what looks good. If you hand the same data table to ten guys and ask them to make a graph, you will get back ten completely different-looking plots.

I have attached examples of what can be done, and I can help you with many aspects of getting plots of data out of EXCEL, but you are the one who ultimately has to choose a presentation style and the detailed data content.

Hopefully, you folks that have stuck with this whole thing are already fluent enough with EXCEL that we do not also have to step through lessons on creating charts, so I am going to skip that step for now. If someone wants help with generating graphs in EXCEL they’re going to have to post or PM me.

View attachment 14_graphs_step 9.xlsx

powderburner said:

... my design undergoes some 44 gs of acceleration at the moment of peak thrust. (Let’s see those high-power guys top THAT!)

Click to expand...

Sure

[youtube]XpfbUuKY_Qg[/youtube]

[youtube]7FBupyMB2CM[/youtube]



(That's a PR 4 inch Nike Smoke on a K2045 Vmax. Around 60G throughout the burn, maybe a bit more. I didn't have an accel onboard, so I'm not completely sure, but the sims indicated around that level of acceleration, and it sure seems believable having seen the flight)

You used a motor that probably cost more than all my rockets put together.

But it sure looked like a great flight!

On a slightly more serious (and relevant) note, Powderburner asked me to look over the calculations and descriptions to double check, and here are my thoughts (mainly for anyone interested in higher-fidelity simulation or the behavior of rockets). They don't really change any of the results, and Powderburner's stuff is all very well done and well documented, but I did have a couple of very minor comments for improved simulations, and about rocket behavior in general:


1) I don't honestly think there's a substantial difference in lift slope between a thin flat plate with square edges and a thin, symmetric airfoil section. They might have slightly different stall behavior, and they definitely have different drag, but I'd bet the stability differences, at least below an angle of attack of 10 degrees or so, would be pretty minimal. In addition, with airfoils as thin as our fins tend to be, I doubt there's much difference between a full airfoil section and simply rounding the leading edge and tapering the trailing edge (with a substantial constant thickness region in the middle). Ditto for beveled edges on a supersonic rocket, so long as the bevels are sufficiently shallow (obviously, the faster the rocket is going, the more it benefits from a more shallow bevel).

2) You can model the atmospheric density changes with reasonable accuracy (and no standard atmosphere lookup or table interpolation needed) by using a simple exponential model. As pointed out earlier in the thread, it isn't really needed for this kind of altitude, but if you want a simple model that will improve the accuracy for higher altitude flights a bit, the exponential model is a great choice. I've used this model with great success for post-flight data analysis many times for flights up to about 10k feet, and it's more than good enough for most uses so long as you don't start getting out of the troposphere. The model I use is relatively straightforward: rho = rho_0*exp(-z/H) in which rho_0 is the density at z = 0 (typically 1.225 kg/m^3, assuming z is measured from sea level), z is the height, and H is the scale height of the atmosphere. For the lower atmosphere of earth, H is about 8.5 km. It's not as good as a true standard atmosphere lookup, but it's not far off either, and it's more than good enough for an approximation (and, needless to say, it's a substantial improvement over the constant-density model if your rocket goes more than perhaps a thousand feet).

3) The data I've seen indicates that the Cd during burn can be substantially different than the Cd post burn. However, this data is pretty much exclusively from large, powerful rockets (>J powered, and in several cases, L+ powered), so I don't know how this applies to smaller rockets. I've never done a lot of engineering or measurement of small rockets, so the constant-CD assumption used here is probably as good as any for now. It certainly wouldn't be worth bothering with any more detailed model until there was some data to support it, as well as a more accurate initial value (since all you can really do with Cd with the spreadsheet is guess, and there's not much of a better way without some pretty complex modeling).


Hopefully this helps some people out, and if anyone has questions about any of it (or some of the more advanced details about the flow around or behavior of a rocket), feel free to ask away. I've spent the last several years of my life working towards the hardest option Powderburner mentioned for determining rocket performance (specifically, a degree in aerospace engineering), and I love discussing this kind of thing in detail.

And thanks for taking the time to check this stuff over.

cjl adds some very good observations about my assumptions and simplifications, and adds a good deal of helpful info for those who might want to widen these sorts of spreadsheets to higher impulse classes.

And that drag coefficient business is like quicksand, you could spend your whole life wading around in there and still not come up with a good answer. My number was exactly that, just a guess, and in my experience you can spend a lot of time trying to reconcile these guesses with tracked flight performance. If anyone thinks I spent too much time posting stuff about performance spreadsheets, we could have posted ten times this much just trying to estimate drag coefficients. Which is why I didn't.

Thanks again, cjl, I appreciate the help!

powderburner said:

And that drag coefficient business is like quicksand, you could spend your whole life wading around in there and still not come up with a good answer. My number was exactly that, just a guess, and in my experience you can spend a lot of time trying to reconcile these guesses with tracked flight performance. If anyone thinks I spent too much time posting stuff about performance spreadsheets, we could have posted ten times this much just trying to estimate drag coefficients. Which is why I didn't.

Click to expand...

Absolutely agreed. Even with some of the more expensive and computationally intensive software packages out there, I'm becoming more and more convinced that drag analysis and prediction, at least at this kind of level, is 30% engineering, 60% luck, and 10% voodoo. For the most part, for amateur rocketry, the best way is to simply guess and then compare with known flight results, adjusting the guess until the simulation agrees with the measured performance. A wind tunnel would be great too, but it's kind of expensive and difficult to obtain for most people.

That much engineering?

Well...

I might have overstated it a bit

Just for grins, I modified the excel spreadsheet using a variable atmospheric density, rather than a constant one as in the spreadsheet above. This was done with the equation from my post above, using an exponential model based on a scale height of 8.5 km. This made a difference in projected altitude for this model of 23 cm. This shows just how irrelevant it is to a low altitude, low powered model. For a higher performing rocket however, this could become substantially more significant. The new spreadsheet is attached.

View attachment 14_graphs_step10.xlsx

I would like to point out one more time to everyone that I certainly did NOT present my spreadsheets as the final answer, as the best there could be. I think I have pointed out the places where I made shortcuts and simplifying assumptions and cjl has posted some excellent ideas on how to continue expanding and improving this sort of calc. I am not a fan of "bells and whistles" for their own sake, but these extra features (like a better atmosphere/density model) will definitely come in handy for just about anything beyond low-power rocketry.

If someone has another spreadsheet that they would like to show, or any other kind of simulation, please please please post it. The whole point of this was to put example(s) out in the open to show other guys how they could go about setting up their own. There are a bujillion ways to do this and I have only shown one. I know there are fresher gray-cells out there than mine, so let's hear from you!

And thanks again, cjl!